Symmetric Matrix

Q.

If A is a square matrix $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$ then $A+A^T$ is a symmetric matrix.

1. True

2. False

Q. Let $A$ and $B$ be two symmetric matrices of same order. If $X=AB+BA$ and $Y=AB-BA,$ then $(XY{)}^T$ is equal to

1. $YX$
2. $-XY$
3. $XY$
4. $-YX$

Q. If A is a square matrix $A+A^T$ is symmetric matrix, then
$A-A^T=$

1. Unit matrix
2. Symmetric matrix
3. Skew symmetric matrix
4. Zero matrix

Q.

A square matrix A is said to be a symmetric matrix if

1. $A=-A^T$

2. $A=A^T$

3. $A=\overline{A}$

4. $A=-\overline{A}$

Q. If A and B are two non singular matrices and both are symmetric and commute each other then

1. Both $A^{-1}B$ and $A^{-1}{B}^{-1}$ are symmetric
2. $A^{-1}B$ is symmetric but $A^{-1}{B}^{-1}$ is not symmetric
3. $A^{-1}{B}^{-1}$ is symmetric but $A^{-1}B$ is not symmetric
4. Neither $A^{-1}B$ nor $A^{-1}{B}^{-1}$ are symmetric

Q. If A and B are two matrices of order '3' such that $3A+4B{B}^T=I$ and $B^{-1}=A^T$, then identify which of the following statements is/are correct?

1. $Tr(A^{-1}-4B^3-BA+I)=9$
2. $Tr(A^{-1}-4B^3+BA)=12$
3. $|A^2-3A^3|=64$
4. $|A^{-1}-4B^3-BA|=8$

Q. Assertion $\left(A\right)$:
The matrix $A=\left[\begin{array}{ccc}0& a& b\\ -a& 0& c\\ -b& -c& 0\end{array}\right]$ is a skew-symmetric matrix.
Reason $\left(R\right)$:
A square matrix $A=a_{ij}$ of order $m$ is said to be skew symmetric if $A^T=-A$.

1. Both $\left(A\right)$ and $\left(R\right)$ individually true but $\left(R\right)$ is not the correct explanation of $\left(A\right)$
2. $\left(A\right)$ is true but $\left(R\right)$ is false
3. $\left(A\right)$ is false but $\left(R\right)$ is true
4. Both $\left(A\right)$ and $\left(R\right)$ individually true and $\left(R\right)$ is correct explanation of $\left(A\right)$

Q. If $A$ is a symmetric matrix and $B$ is a skew- symmetrix matrix such that $A+B=\left[\begin{array}{cc}2& 3\\ 5& -1\end{array}\right],$ then $AB$ is equal to :

1. $\left[\begin{array}{cc}-4& 2\\ 1& 4\end{array}\right]$
2. $\left[\begin{array}{cc}4& -2\\ 1& -4\end{array}\right]$
3. $\left[\begin{array}{cc}4& -2\\ -1& -4\end{array}\right]$
4. $\left[\begin{array}{cc}-4& -2\\ -1& 4\end{array}\right]$

Q. Out of the following which is a skew- symmetric matrix

1. $\left[\begin{array}{ccc}0& 4& 5\\ -4& 0& -6\\ -5& 6& 0\end{array}\right]$
2. $\left[\begin{array}{ccc}1& 4& 5\\ -4& 1& -6\\ -5& 6& 1\end{array}\right]$
3. $\left[\begin{array}{ccc}1& 4& 5\\ -4& 2& -6\\ -5& 6& 3\end{array}\right]$
4. $\left[\begin{array}{ccc}i+1& 4& 5\\ -4& i& -6\\ -5& 6& i\end{array}\right]$

Q. Let $A=\left[\begin{array}{ccc}0& 2& -3\\ -2& 0& 6\\ 3& -6& 0\end{array}\right],$ then which of the following is/are correct?

माना $A=\left[\begin{array}{ccc}0& 2& -3\\ -2& 0& 6\\ 3& -6& 0\end{array}\right],$ तब निम्नलिखित में से कौनसा/कौनसे विकल्प सही है/हैं?

1. det(A) = 0
2. A is a symmetric matrix
   
   A एक सममित आव्यूह है
3. A is a skew-symmetric matrix
   
   A एक विषम सममित आव्यूह है
4. det(A) is a prime number
   
   det(A) अभाज्य संख्या है